A polymatroid approach to separable convex optimization with linear ascending constraints

Abstract

We revisit a problem studied by Padakandla and Sundaresan [SIAM J. Optim., August 2009] on the minimization of a separable convex function subject to linear ascending constraints. The problem arises as the core optimization in several resource allocation problems in wireless communication settings. It is also a special case of an optimization of a separable convex function over the bases of a specially structured polymatroid. We give an alternative proof of the correctness of the algorithm of Padakandla and Sundaresan. In the process we relax some of their restrictions placed on the objective function.

Convex Separable Problems With Linear Constraints in Signal Processing and Communications

“…Furthermore, differently from [18] we do not impose any constraints on the slopes of f n . It is also worth mentioning that at the time of submission we became aware (through a private correspondence with the authors) of [20] in which the problem originally solved in [18] has been revisited in light of the theory of polymatroids. In particular, in [20] the authors have removed some of the restrictions on functions f n that were present in [18].…”

“…It is also worth mentioning that at the time of submission we became aware (through a private correspondence with the authors) of [20] in which the problem originally solved in [18] has been revisited in light of the theory of polymatroids. In particular, in [20] the authors have removed some of the restrictions on functions f n that were present in [18]. This allows them to come up with a solution similar to the one we propose in this work.…”

Algorithms for separable convex optimization with linear ascending constraints

“…Let w e : [0, b e ) → R, e = 1, 2, · · · , n be convex functions where 0 < b e ≤ ∞ and R = R ∪ {−∞, +∞} be the extended real line. We wish to minimize a separable objective function W : R n → R as in Problem Π : Minimize W (x) := n e=1 w e (x(e)) (1) subject to x(e) ∈ [0, β(e)], e = 1, 2, · · · , n,…”

Convex separable problems with linear and box constraints

“…It is also worth mentioning that at the time of submission we became aware of[13] in which the authors come up with an extended solution much similar to the proposed one using the theory of polymatroids.…”